Exam 7: Area of a Region Between Two Curves

An open tank has the shape of a right circular cone. The tank is 9 feet across the top and 8 feet high. How much work is done in emptying the tank by pumping the water over the top Edge? Note: The density of water is 62.4 lbs per cubic foot.

A barn is 75 feet long and 50 feet wide. A cross section of the roof is the inverted catenary $y = 41 - 15 \left( e ^ { \frac { x } { 30 } } + e ^ { - \frac { x } { 30 } } \right)$ . Find the number of square feet of roofing on the barn. Round your answer to the nearest integer.

A quantity of gas with an initial volume of 8 cubic feet and a pressure of 1500 pounds per square foot expands to a volume of 10 cubic feet. Find the work done by the gas for the Given volume and pressure. Round your answer to two decimal places.

The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form.

$\text { An ornamental light bulb is designed by revolving the graph of } y = \frac { 1 } { 3 } x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } }$ about the x-axis where x and y are measured in feet. Approximate the amount of glass needed to make the bulb. (Assume that the glass is 0.016 inches thick.) Round your answer to four decimal places.

$\text { Find the volume of the solid generated by rotating the circle } x ^ { 2 } + ( y - 10 ) ^ { 2 } = 64$ about the x-axis.

Use the shell method to find the volume of the solid generated by revolving the plane region about the line $x = 12$ . $y = \sqrt [ 3 ] { x } , y = 0 , x = 8$

Find the area of the region bounded by equations by integrating (i) with respect to x and (ii) with respect to y. y= y=72-x

$\text { Find the arc length of the graph of the function } y = \frac { 3 } { 2 } x ^ { \frac { 2 } { 3 } } + 10 \text { over the interval }$ [1,64].

Find the arc length of the graph of the function $x = \frac { 1 } { 3 } ( y - 3 ) \sqrt { y } \quad \text {over the interval }$ $1 \leq y \leq 144$

The chief financial officer of a company reports that profits for the past fiscal year were $ million. The officer predicts that profits for the next 7 years will grow at a continuous annual rate somewhere between $4 \frac { 1 } { 2 } \%$ and $6 \%$ . Estimate the cumulative difference in total profit over the 7 years based on the predicted range of growth rates. Round your answer to three decimal places.

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by $y = 8$ and $y = 16 - \frac { x ^ { 2 } } { 16 }$ about the $x$ -axis.

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis. $y = 10 - x , y = 0 , x = 0$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x \text {-axis. }$ $y = \frac { 1 } { x } , y = 0 , x = 8 , x = 10$

Use the disk or shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y = 22$ . $y = \frac { 22 } { x ^ { 2 } } , y = 0 , x = 1 , x = 11$

A cylindrical water tank 5 meters high with a radius of 2 meters is buried so that the top of the tank is 1 meter below ground level. How much work is done in pumping a full tank of Water up to ground level? (The water weighs 9800 newtons per cubic meter.)

A lunar module weighs 16 tons on the surface of Earth. How much work is done in propelling the module from the surface of the moon to a height of 50 miles? Consider the radius of The moon to be 1100 miles and its force of gravity to be one-sixth that of Earth.

A 17-foot chain that weighs 4 pounds per cubic foot hangs from a winch 17 feet above ground level. Find the work done by the winch in winding up the entire chain.

A force of 10 pounds compresses a 21-inch spring 8 inches. How much work is done in compressing the spring from a length of 15 inches to a length of 13 inches?

Find the area of the surface generated by revolving the curve about the x-axis. $y = \frac { 1 } { 7 } x ^ { 3 } , 0 \leq x \leq 7$